Statistics with Economics and Business Applications

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Statistics with Economics and Business
Applications
Chapter 6 Sampling Distributions Random Sample,
Central Limit Theorem
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Review

• I. Whats in last lecture?
• Normal Probability Distribution. Chapter
  5
• II. What's in this lecture?
• Random Sample, Central Limit Theorem Read
  Chapter 6

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Introduction

• Parameters are numerical descriptive measures for
  populations.
• Two parameters for a normal distribution mean m
  and standard deviation s.
• One parameter for a binomial distribution the
  success probability of each trial p.
• Often the values of parameters that specify the
  exact form of a distribution are unknown.
• You must rely on the sample to learn about these
  parameters.

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Sampling

• Examples
• A pollster is sure that the responses to his
  agree/disagree question will follow a binomial
  distribution, but p, the proportion of those who
  agree in the population, is unknown.
• An agronomist believes that the yield per acre of
  a variety of wheat is approximately normally
  distributed, but the mean m and the standard
  deviation s of the yields are unknown.
• If you want the sample to provide reliable
  information about the population, you must select
  your sample in a certain way!

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Simple Random Sampling

• The sampling plan or experimental design
  determines the amount of information you can
  extract, and often allows you to measure the
  reliability of your inference.
• Simple random sampling is a method of sampling
  that allows each possible sample of size n an
  equal probability of being selected.

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Sampling Distributions

• Any numerical descriptive measures calculated
  from the sample are called statistics.
• Statistics vary from sample to sample and hence
  are random variables. This variability is called
  sampling variability.
• The probability distributions for statistics are
  called sampling distributions.
• In repeated sampling, they tell us what values
  of the statistics can occur and how often each
  value occurs.

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Example
Population 3, 5, 2, 1 Draw samples of size n 3
without replacement
Each value of x-bar is equally likely, with
probability 1/4
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Example

• Consider a population that consists of the
  numbers 1, 2, 3, 4 and 5 generated in a manner
  that the probability of each of those values is
  0.2 no matter what the previous selections were.
  This population could be described as the outcome
  associated with a spinner such as given below
  with the distribution next to it.

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Example

• If the sampling distribution for the means of
  samples of size two is analyzed, it looks like

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Example

• The original distribution and the sampling
  distribution of means of samples with n2 are
  given below.

Original distribution
Sampling distribution n 2
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Example

• Sampling distributions for n3 and n4 were
  calculated and are illustrated below. The shape
  is getting closer and closer to the normal
  distribution.

Original distribution
Sampling distribution n 2
Sampling distribution n 3
Sampling distribution n 4
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Sampling Distribution of
If a random sample of n measurements is selected
from a population with mean m and standard
deviation s, the sampling distribution of the
sample mean will have a mean
and a standard deviation
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Why is this Important?
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How Large is Large?
If the sample is normal, then the sampling
distribution of will also be normal, no
matter what the sample size. When the sample
population is approximately symmetric, the
distribution becomes approximately normal for
relatively small values of n. When the sample
population is skewed, the sample size must be at
least 30 before the sampling distribution of
becomes approximately normal.
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Illustrations of Sampling Distributions
Symmetric normal like population
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Illustrations of Sampling Distributions
Skewed population
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Finding Probabilities for the Sample Mean
Example A random sample of size n 16 from a
normal distribution with m 10 and s 8.
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Example
A soda filling machine is supposed to fill cans
of soda with 12 fluid ounces. Suppose that the
fills are actually normally distributed with a
mean of 12.1 oz and a standard deviation of .2
oz. The probability of one can less than 12 is
What is the probability that the average fill for
a 6-pack of soda is less than 12 oz?
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The Sampling Distribution of the Sample Proportion
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The Sampling Distribution of the Sample Proportion
The standard deviation of p-hat is sometimes
called the STANDARD ERROR (SE) of p-hat.
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Finding Probabilities for the Sample Proportion
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Example
The soda bottler in the previous example claims
that only 5 of the soda cans are underfilled.
A quality control technician randomly samples
200 cans of soda. What is the probability that
more than 10 of the cans are underfilled?
n 200 S underfilled can p P(S) .05 q
.95 np 10 nq 190
This would be very unusual, if indeed p .05!
OK to use the normal approximation
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Example
Suppose 3 of the people contacted by phone are
receptive to a certain sales pitch and buy your
product. If your sales staff contacts 2000
people, what is the probability that more than
100 of the people contacted will purchase your
product?
OK to use the normal approximation
n2000, p 0.03, np60, nq1940,
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Key Concepts

• I. Sampling Plans and Experimental Designs
• Simple random sampling Each possible sample
  is equally likely to occur.
• II. Statistics and Sampling Distributions
• 1. Sampling distributions describe the possible
  values of a statistic and how often they occur in
  repeated sampling.
• 2. The Central Limit Theorem states that sums and
  averages of measurements from a nonnormal
  population with finite mean m and standard
  deviation s have approximately normal
  distributions for large samples of size n.

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Key Concepts

• III. Sampling Distribution of the Sample Mean
• 1. When samples of size n are drawn from a normal
  populationwith mean m and variance s 2, the
  sample mean has a normal distribution with
  mean m and variance s 2/n.
• 2. When samples of size n are drawn from a
  nonnormal population with mean m and variance s
  2, the Central Limit Theorem ensures that the
  sample mean will have an approximately normal
  distribution with mean m and variances 2 /n when
  n is large (n ³ 30).
• 3. Probabilities involving the sample mean m can
  be calculatedby standardizing the value of
  using

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Key Concepts

• IV. Sampling Distribution of the Sample
  Proportion
• When samples of size n are drawn from a binomial
  population with parameter p, the sample
  proportion will have an approximately normal
  distribution with mean p and variance pq /n as
  long as np gt 5 and nq gt 5.
• 2. Probabilities involving the sample proportion
  can be calculated by standardizing the value
  using